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# Section2.3posted - STOR 155 Section 2 Thursday Section 2.3...

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STOR 155, Section 2 Thursday, January 28, 2010 Section 2.3

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Section 2.3: Least-squares regression If there’s a linear relationship between an explanatory variable x and a response variable y , we would like a straight line that expresses the relationship. A straight line in the xy- plane has an equation y = b 0 + b 1 x. When x = 0 , y = b 0 . b 0 is the intercept . When x = 1, y = b 0 + b 1 . Any increase of 1 unit in x produces an increase of b 1 units in y . b 1 is the slope . Equation of line: y = b 0 + b 1 x x = 0, y = b 0 x = 1, y = b 0 +b 1 | x=1 x=0
Some straight lines and their equations y = 2 + 1.4 x y = −3 + 2 x y = 1 − 0.5 x y = 2 (i.e. y = 2 + 0 x ) x x x x y y y y

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Least-squares regression What’s the best line to fit the points in a scatterplot? By general agreement, it’s the line that makes the sum of the squares of those vertical distances as small as possible. That is, the least-squares regression line . ) How to find the equation of that line: 3 slides below. Why do we call it a regression line? 8 slides below.
x b b y 1 0 ˆ + = The equation of the least- squares regression line is written For any x , ŷ is the value above that x on the line, not a value of y from the dataset. ŷ is the predicted value of y for that value of x, not the observed value. The least-squares regression line is a predictor. Least-squares regression line (For b 0 , say “ b -zero” or “ b -nought.”) 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 10 x y x=5, y=7 x=5, ŷ=4.3 Dataset has 3 points ( x,y ) (blue). For every x , there is a predicted value ŷ given by the line.

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Fat-gain study Rresearchers put subjects on a controlled high-fat diet, and they measured x (amount of non- exercise activity, in calories) and y (amount of fat gained). The regression line is shown.
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Section2.3posted - STOR 155 Section 2 Thursday Section 2.3...

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